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Andreas Scherer 87bf61ced3 Fix typos. (#88) 
* Fix typos.

A quick check with 'aspell -c -l en_US'.

* Fix some more typos.

* More typos fixed.
2017-05-26 13:45:40 -06:00

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<HTML>
<!--
Copyright (c) Jeremy Siek 2000
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at
http://www.boost.org/LICENSE_1_0.txt)
-->
<Head>
<Title>Boost Graph Concepts</Title>
<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
ALINK="#ff0000">
<IMG SRC="../../../boost.png"
ALT="C++ Boost" width="277" height="86">
<BR Clear>
<H1><A NAME="chapter:graph-concepts"></A>
Graph Concepts
</H1>
<P>
The heart of the Boost Graph Library (BGL) is the interface, or
concepts (in the parlance of generic programming), that define how a
graph can be examined and manipulated in a data-structure neutral
fashion. In fact, the BGL interface need not even be implemented using
a data-structure, as for some problems it is easier or more efficient
to define a graph implicitly based on some functions.
<P>
The BGL interface does not appear as a single graph concept. Instead
it is factored into much smaller pieces. The reason for this is that
the purpose of a concept is to summarize the requirements for
<i>particular</i> algorithms. Any one algorithm does not need every
kind of graph operation, typically only a small subset. Furthermore,
there are many graph data-structures that can not provide efficient
implementations of all the operations, but provide highly efficient
implementations of the operations necessary for a particular algorithm.
By factoring the graph interface into many smaller concepts we
provide the graph algorithm writer with a good selection from which to
choose the concept that is the closest match for their algorithm.
Note that because of the use of traits classes rather than member
types, it is not safe (and often will not work) to define subclasses of BGL
graph types; those types may be missing important traits and properties that
were defined externally to the class definition.
<H2>Graph Structure Concepts Overview</H2>
<P>
<A HREF="#fig:graph-concepts">Figure 1</A> shows the refinements
relations between the graph concepts. The reason for factoring the
graph interface into so many concepts is to encourage algorithm
interfaces to require and use only the minimum interface of a graph,
thereby increasing the reusability of the algorithm.
<p></p>
<DIV ALIGN="CENTER"><A NAME="fig:graph-concepts"></A></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1:</STRONG>
The graph concepts and refinement relationships.
</CAPTION>
<TR><TD><IMG SRC="./figs/concepts.gif"></TD></TR>
</TABLE>
</DIV>
<p></p>
<A HREF="#tab:graph-concept-reqs">Table&nbsp;1</A>
gives a summary of the valid expressions and associated types for the
graph concepts and provides links to the detailed descriptions of
each of the concepts. The notation used in the table is as follows.
<h3>Notation</h3>
<Table>
<TR>
<TD><tt>G</tt></TD>
<TD>A type that is a model of Graph.</TD>
</TR>
<TR>
<TD><tt>g</tt></TD>
<TD>An object of type <tt>G</tt>.</TD>
</TR>
<TR>
<TD><tt>e</tt></TD>
<TD>An object of type <tt>boost::graph_traits&lt;G&gt;::edge_descriptor</tt>.</TD>
</TR>
<TR>
<TD><tt>e_iter</tt></TD>
<TD>An object of type <tt>boost::graph_traits&lt;G&gt;::out_edge_iterator</tt>.</TD>
</TR>
<TR>
<TD><tt>u,v</tt></TD>
<TD>Are objects of type <tt>boost::graph_traits&lt;G&gt;::vertex_descriptor</tt>.</TD>
</TR>
<TR>
<TD><TT>ep</TT></TD><TD>is an object of type <TT>G::edge_property_type</TT></TD>
</TR>
<TR>
<TD><TT>vp</TT></TD><TD>is an object of type <TT>G::vertex_property_type</TT></TD>
</TR>
<TR>
<TD><tt>Property</tt></TD>
<TD>A type used to specify a vertex or edge property.</TD>
</TR>
<TR>
<TD><tt>property</tt></TD>
<TD>An object of type <tt>Property</tt>.</td>
</TR>
</table>
<P>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="tab:graph-concept-reqs"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Table 1:</STRONG>
Summary of the graph concepts.
</CAPTION>
<TR><TD>
<TABLE border>
<TR><TH ALIGN="LEFT">
<B>Expression</B> </TH>
<TH ALIGN="LEFT" VALIGN="TOP"> <B>Return Type or Description</B> </TH>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=2>
<a href="./Graph.html">Graph</a> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::vertex_descriptor</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> The type for
vertex representative objects. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::edge_descriptor</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> The type for
edge representative objects. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::directed_category</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> Directed or undirected? </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::edge_parallel_category</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> Allow parallel edges? </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::traversal_category</TT> </TD> <TD
ALIGN="LEFT" VALIGN="TOP">The ways in which the vertices and edges of
the graph can be visited.</TD>
</TR>
<!---------------------------------------------------------------->
<TR><TD ALIGN="LEFT" COLSPAN=2>
<a href="./IncidenceGraph.html">IncidenceGraph</a> refines Graph </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::out_edge_iterator</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> Iterate through
the out-edges. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::degree_size_type</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> The integer type for
vertex degree. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>out_edges(v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>std::pair&lt;out_edge_iterator, out_edge_iterator&gt;</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>source(e, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>vertex_descriptor</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>target(e, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>vertex_descriptor</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>out_degree(v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>degree_size_type</TT> </TD>
</TR>
<!---------------------------------------------------------------->
<TR><TD ALIGN="LEFT" COLSPAN=2>
<a href="./BidirectionalGraph.html">BidirectionalGraph</a> refines
IncidenceGraph </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::in_edge_iterator</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> Iterate through the in-edges. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>in_edges(v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>std::pair&lt;in_edge_iterator, in_edge_iterator&gt;</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>in_degree(v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>degree_size_type</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>degree(e, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>degree_size_type</TT> </TD>
</TR>
<!---------------------------------------------------------------->
<TR><TD ALIGN="LEFT" COLSPAN=2>
<a href="./AdjacencyGraph.html">AdjacencyGraph</a> refines Graph</TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::adjacency_iterator</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> Iterate through
adjacent vertices. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>adjacent_vertices(v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"><TT>std::pair&lt;adjacency_iterator, adjacency_iterator&gt;</TT> </TD>
</TR>
<!---------------------------------------------------------------->
<TR><TD ALIGN="LEFT" COLSPAN=2>
<a href="./VertexListGraph.html">VertexListGraph</a> refines
Graph</TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::vertex_iterator</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> Iterate through the
graph's vertex set. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::vertices_size_type</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> The unsigned integer type for
number of vertices in the graph. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>vertices(g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"><TT>std::pair&lt;vertex_iterator, vertex_iterator&gt;</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>num_vertices(g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>vertices_size_type</TT> </TD>
</TR>
<!---------------------------------------------------------------->
<TR><TD ALIGN="LEFT" COLSPAN=2>
<a href="./EdgeListGraph.html">EdgeListGraph</a> refines Graph</TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::edge_iterator</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> Iterate through the graph's
edge set. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::graph_traits&lt;G&gt;::edges_size_type</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> The unsigned integer type for
number of edges in the graph. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>edges(g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>std::pair&lt;edge_iterator, edge_iterator&gt;</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>num_edges(g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>edges_size_type</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>source(e, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>vertex_descriptor</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>target(e, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>vertex_descriptor</TT> </TD>
</TR>
<!---------------------------------------------------------------->
<TR><TD ALIGN="LEFT" COLSPAN=2>
<a href="./AdjacencyMatrix.html">AdjacencyMatrix</a> refines Graph</TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>edge(u, v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>std::pair&lt;edge_descriptor, bool&gt;</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT" COLSPAN=2>
<a href="./MutableGraph.html">MutableGraph</a> refines
Graph</TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>add_vertex(g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>vertex_descriptor</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>clear_vertex(v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>void</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>remove_vertex(v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>void</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>add_edge(u, v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>std::pair&lt;edge_descriptor, bool&gt;</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>remove_edge(u, v, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>void</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>remove_edge(e, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>void</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>remove_edge(e_iter, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>void</TT> </TD>
</TR>
<!---------------------------------------------------------------->
<TR><TD ALIGN="LEFT" COLSPAN=2>
<a href="./MutablePropertyGraph.html">MutablePropertyGraph</a> refines
Graph</TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>add_vertex(vp, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>vertex_descriptor</TT> </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>add_edge(u, v, ep, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> <TT>std::pair&lt;edge_descriptor,
bool&gt;</TT> </TD>
</TR>
<!---------------------------------------------------------------->
<TR>
<TD ALIGN="LEFT" COLSPAN=2>
<a href="./PropertyGraph.html">PropertyGraph</a> refines Graph</TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::property_map&lt;G, Property&gt;::type</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP">Type for a mutable property map.</TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>boost::property_map&lt;G, Property&gt;::const_type</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP">Type for a non-mutable property map.</TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>get(property, g)</TT> </TD>
<TD ALIGN="LEFT" VALIGN="TOP"> Function to get a property map. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>get(property,&nbsp;g,&nbsp;x)</TT>
</TD>
<TD ALIGN="LEFT" VALIGN="TOP">Get property value for vertex or edge <tt>x</tt>. </TD>
</TR>
<TR><TD ALIGN="LEFT">
<TT>put(property,&nbsp;g,&nbsp;x,&nbsp;v)</TT>
</TD>
<TD ALIGN="LEFT" VALIGN="TOP">Set property value for vertex or edge
<tt>x</tt> to <tt>v</tt>. </TD>
</TR>
</table>
</table>
</DIV><P></P>
<BR>
<P>
<H2><A NAME="sec:undirected-graphs"></A>
Undirected Graphs
</H2>
<P>
The interface that the BGL provides for accessing and manipulating an
undirected graph is the same as the interface for directed graphs
described in the following sections, however there are some
differences in the behaviour and semantics. For example, in a
directed graph we can talk about out-edges and in-edges of a vertex.
In an undirected graph there is no ``in'' and ``out'', there are just
edges incident to a vertex. Nevertheless, in the BGL we still use the
<TT>out_edges()</TT> function (or <TT>in_edges()</TT>) to access the
incident edges in an undirected graph. Similarly, an undirected edge
has no ``source'' and ``target'' but merely an unordered pair of
vertices, but in the BGL we still use <TT>source()</TT> and
<TT>target()</TT> to access these vertices. The reason the BGL does
not provide a separate interface for undirected graphs is that many
algorithms on directed graphs also work on undirected graphs, and it
would be inconvenient to have to duplicate the algorithms just because
of an interface difference. When using undirected graphs just mentally
disregard the directionality in the function names. The example below
demonstrates using the <TT>out_edges()</TT>, <TT>source()</TT>, and
<TT>target()</TT> with an undirected graph. The source code for this
example and the following one can be found in <a
href="../example/undirected_adjacency_list.cpp"><TT>example/undirected_adjacency_list.cpp</TT></a>.
<P>
<PRE>
const int V = 2;
typedef ... UndirectedGraph;
UndirectedGraph undigraph(V);
std::cout &lt;&lt; "the edges incident to v: ";
boost::graph_traits&lt;UndirectedGraph&gt;::out_edge_iterator e, e_end;
boost::graph_traits&lt;UndirectedGraph&gt;::vertex_descriptor
s = vertex(0, undigraph);
for (boost::tie(e, e_end) = out_edges(s, undigraph); e != e_end; ++e)
std::cout &lt;&lt; "(" &lt;&lt; source(*e, undigraph)
&lt;&lt; "," &lt;&lt; target(*e, undigraph) &lt;&lt; ")" &lt;&lt; endl;
</PRE>
<P>
Even though the interface is the same for undirected graphs, there are
some behavioral differences because edge equality is defined
differently. In a directed graph, edge <i>(u,v)</i> is never equal to edge
<i>(v,u)</i>, but in an undirected graph they may be equal. If the
undirected graph is a multigraph then <i>(u,v)</i> and <i>(v,u)</i> might be
parallel edges. If the graph is not a multigraph then <i>(u,v)</i> and
<i>(v,u)</i> must be the same edge.
<P>
In the example below the edge equality test will return <TT>false</TT>
for the directed graph and <TT>true</TT> for the undirected graph. The
difference also affects the meaning of <TT>add_edge()</TT>. In the
example below, if we had also written <TT>add_edge(v, u,
undigraph)</TT>, this would have added a parallel edge between
<i>u</i> and <i>v</i> (provided the graph type allows parallel
edges). The difference in edge equality also affects the association
of edge properties. In the directed graph, the edges <i>(u,v)</i> and
<i>(v,u)</i> can have distinct weight values, whereas in the
undirected graph the weight of <i>(u,v)</i> is the same as the weight
of <i>(v,u)</i> since they are the same edge.
<P>
<PRE>
typedef ... DirectedGraph;
DirectedGraph digraph(V);
{
boost::graph_traits&lt;DirectedGraph&gt;::vertex_descriptor u, v;
u = vertex(0, digraph);
v = vertex(1, digraph);
add_edge(digraph, u, v, Weight(1.2));
add_edge(digraph, v, u, Weight(2.4));
boost::graph_traits&lt;DirectedGraph&gt;::edge_descriptor e1, e2;
bool found;
boost::tie(e1, found) = edge(u, v, digraph);
boost::tie(e2, found) = edge(v, u, digraph);
std::cout &lt;&lt; "in a directed graph is ";
std::cout &lt;&lt; "(u,v) == (v,u) ? " &lt;&lt; (e1 == e2) &lt;&lt; std::endl;
property_map&lt;DirectedGraph, edge_weight_t&gt;::type
weight = get(edge_weight, digraph);
cout &lt;&lt; "weight[(u,v)] = " &lt;&lt; get(weight, e1) &lt;&lt; endl;
cout &lt;&lt; "weight[(v,u)] = " &lt;&lt; get(weight, e2) &lt;&lt; endl;
}
{
boost::graph_traits&lt;UndirectedGraph&gt;::vertex_descriptor u, v;
u = vertex(0, undigraph);
v = vertex(1, undigraph);
add_edge(undigraph, u, v, Weight(3.1));
boost::graph_traits&lt;UndirectedGraph&gt;::edge_descriptor e1, e2;
bool found;
boost::tie(e1, found) = edge(u, v, undigraph);
boost::tie(e2, found) = edge(v, u, undigraph);
std::cout &lt;&lt; "in an undirected graph is ";
std::cout &lt;&lt; "(u,v) == (v,u) ? " &lt;&lt; (e1 == e2) &lt;&lt; std::endl;
property_map&lt;UndirectedGraph, edge_weight_t&gt;::type
weight = get(edge_weight, undigraph);
cout &lt;&lt; "weight[(u,v)] = " &lt;&lt; get(weight, e1) &lt;&lt; endl;
cout &lt;&lt; "weight[(v,u)] = " &lt;&lt; get(weight, e2) &lt;&lt; endl;
}
</PRE>
The output is:
<PRE>
in a directed graph is (u,v) == (v,u) ? 0
weight[(u,v)] = 1.2
weight[(v,u)] = 2.4
in an undirected graph is (u,v) == (v,u) ? 1
weight[(u,v)] = 3.1
weight[(v,u)] = 3.1
</PRE>
<br>
<HR>
<TABLE>
<TR valign=top>
<TD nowrap>Copyright &copy; 2000-2001</TD><TD>
<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
</TD></TR></TABLE>
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